Versal deformations and versality in central extensions of Jacobi schemes

Abstract

Let $\mathfrak{L}_m$ be the scheme of the laws defined by the Jacobi identities on $\mathbb{K}^m$ with $\mathbb{K}$ a field. A deformation of $\mathfrak{g} \in \mathfrak{L}_m$, parametrized by a local ring A, is a local morphism from the local ring of $\mathfrak{L}_m$ at ϕ _m to A. The problem of classifying all the deformation equivalence classes of a Lie algebra with given base is solved by “versal” deformations. First, we give an algorithm for computing versal deformations. Second, we prove there is a bijection between the deformation equivalence classes of an algebraic Lie algebra ϕ _m = R ⋉ φ _n in $\mathfrak{L}_m$ and its nilpotent radical φ _n in the R-invariant scheme $\mathfrak{L}_{n}^{\rm R}$ with reductive part R, under some conditions. So the versal deformations of ϕ _m in $\mathfrak{L}_m$ are deduced from those of φ _n in $\mathfrak{L}_{n}^{\rm R}$, which is a more simple problem. Third, we study versality in central extensions of Lie algebras. Finally, we calculate versal deformations of some Lie algebras.